This is a given recurrence relation
$a_n=19(F_0a_{n-1}+F_1a_{n-2}+...+F_{n-1}a_0)$
where $F_n$ is Fibonacci number and $a_0=9$. Find the generating function $A(x)$ of the sequence $a_n$
I get the generating function for $F_n=\frac x{1-x-x^2}$
I try this for hour with derivative method $A'(x)$ but I still do not get it. Can you help me please, any method is fine.
From $$ F(x)A(x) = \sum_{n=0}^\infty F_nx^n \cdot \sum_{n=0}^\infty a_nx^n = \sum_{n=0}^\infty \Bigl( \sum_{k=0}^n F_k a_{n-k}\Bigr) x^n $$
we see that $F_0a_{n-1}+F_1a_{n-2}+...+F_{n-1}a_0$ is the coefficient of $x^{n-1}$ in $F(x)A(x)\,$, or the coefficient of $x^n$ in $xF(x)A(x)\,$.
Therefore your recurrence relation can be expressed in terms of the generating functions as $$ A(x) = 9 + 19 \, x \, A(x) F(x) $$ and $F(x)$ is known.